Effective potential energy for relativistic particles in thefield of inclined rotating magnetized sphere

V. Epp1,2∗and M. A. Masterova1

1Tomsk State Pedagogical University, 634061 Tomsk, Russia2Tomsk State University, 634050 Tomsk, Russia

Abstract

The dynamics of a charged relativistic particle in electromagnetic field of a rotatingmagnetized celestial body with the magnetic axis inclined to the axis of rotation isstudied. The covariant Lagrangian function in the rotating reference frame is found.Effective potential energy is defined on the base of the first integral of motion. Thestructure of the equipotential surfaces for a relativistic charged particle is studied anddepicted for different values of the dipole moment. It is shown that there are trappingregions for the particles of definite energies.

Keywords: Størmer’s problem, magnetic dipole, equation of motion, magneto-sphere, inclined rotator, potential energy, trapping zones.

1 Introduction

Motion of the charged particles in the field of a magnetized rotating celestial body is oflarge practical significance for astrophysics. For example, a charged particle in the Earthmagnetic field is moving within the closed regions which are named radiation belts [1, 2].The trajectories of a charged particle in the dipolar magnetic field where studied in thepapers [3–5] and [6].

More complicated case is the case when direction of the magnetic moment differs fromdirection of axis of rotation. In this case an electric field is induced inside and outside of thebody. The neutron stars and pulsars are examples of such objects. The first model of electricfield which is generated in the neighbourhood of a neutron star was developed by Deutsch [7].Some other models were suggested and studied by several authors [8]. Most of these modelsare based on assumption that the neutron star is a conducting sphere. Electromagnetic fieldin this case differs essentially from the pure dipole field. Magnetic field of such objects ingood approximation can be described as the field of an inclined rotating magnetized sphereor ”oblique rotator” [9]. Theoretical study of the field of an oblique rotator has a longhistory. We discuss it in more details in Section 2.

∗epp@tspu.edu.ru

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14

Allowed and forbidden regions of the motion of charged particles in such field was studiedby Katsiaris and Psillakis [10]. Dynamics of a charged particle near the force-free surfaceof a rotating magnetized sphere was explored in [11, 12]. Some issues of charged particledynamics within the electromagetic vacuum fields of an inclined rotator have already beendiscussed in the papers [13, 14].

Effective potential energy for a non-relativistic particle in the field of inclined rotatingdipole was investigated in details in our recent paper [15], further referred to as Paper I. Thecalculations were made for the near region, i.e. for distances much less then the radius of thelight cylinder. In the present paper we study the structure of the effective potential energyfor a relativistic particle in the field of inclined rotating magnetized sphere at the distancesup to the light cylinder.

In Section 2 we show that the field of such sphere calculated by different authors underdifferent assumption coincides with the field of rotating magnetic dipole if the sphere is uni-formly magnetized and rotates with a non-relativistic speed. In Section 3 we present therelativistic Lagrange function for a charged particle in the arbitrary, uniformly rotating elec-tromagnetic field. The integral of motion of such Lagrange system is calculated. Existenceof the integral of motion gives the possibility to introduce an effective potential energy whichallows studying some general features of the particle motion without solving the equationsof motion. Section 4 represents analysis of the main properties of the effective potentialenergy. The equipotential surfaces are obtained by numerical calculations and demonstratedin pictures of Section 5. Section 7 contains discussion of the results and our conclusions.

2 The electromagnetic field of a rotating magnetized

sphere

In this section we analyse the field of rotating magnetized sphere. There are different waysof modelling the field of an oblique rotator.

Deutsch describes a non-relativistic rotating magnetized star as a perfectly conductingsphere in rigid rotation in vacuo [7]. In order to introduce a relativistic model of the fieldsource Belinsky et al. [16,17] considered an infinitely thin permanent magnet of finite length.This model is acceptable for calculation of the field at large distances from the source, butit can not be used for the near field calculations.

In paper [18] has been found an exact special relativistic solution for the electromagneticfield in the interior and exterior of rapidly rotating perfectly conducting magnetized sphere.The calculation of the field is made as generalization of the field of slowly rotating magnetizedneutron star, which was studied in [19] under consideration of general relativistic effects.The field of a rotating magnetized sphere which is neither a conductor nor a dielectricwas calculated by Kaburaki [20]. There is a great variety of other papers which presentcalculations of the electromagnetic field of rotating magnetized sphere – see references inthe articles cited above. The results differs essentially dependent on the used model of themagnetized sphere and its speed of rotation.

The model of relativistically rotating sphere is rather complicate. First of all, a solidsphere is incompatible with the theory of relativity. Hence, we have to consider a liquidmodel or gaseous. Therefore it is not a sphere. Secondly, the inner field of fast rotating

2

body depends on the form of the body and on the nature of magnetization. In this paper weaccept as a model of the exterior electromagnetic field the field of slowly rotating uniformlymagnetized sphere. We show that in the non-relativistic limit the results of many authorscited above give the same field.

Let us start with the expression for the exterior electromagnetic field obtained by Deutsch

[7]. We expand these equations in powers of a =ωr0c

, where ω is the angular speed of

rotation, r0 is the radius of the sphere, and c is the speed of light. But we keep termslike r0/r which are sufficient near the surface of the sphere. Up to the first order of a wereceive the next equations for the electric (E) and magnetic (H) field vectors in a sphericalcoordinate system r, θ, ϕ (axis Z is directed along the vector of angular velocity ω):

Er = −µk3a2

2ρ4[cosα(3 cos 2θ + 1) + sinα sin 2θ(3C − ρ2 cosλ)],

Eθ = −µk3

ρ2

[C sinα

(1− a2

ρ2cos 2θ

)+a2

ρ2cosα sin 2θ

], (1)

Eϕ =µk3

ρ2S sinα cos θ

(1− a2

ρ2

),

Hr =2µk3

ρ3(cosα cos θ + C sinα sin θ),

Hθ =µk3

ρ3[cosα sin θ − sinα cos θ(C − ρ2 cosλ)], (2)

Hϕ =µk3

ρ3sinα(S − ρ2 sinλ) ,

whereS = sinλ− ρ cosλ, C = cosλ+ ρ sinλ,

µ is the dipole moment vector, µ = |µ|, λ = ρ + ϕ − ωt, ρ = rω/c, k = ω/c, and α is theangle between the vectors µ and ω. We have also expanded:

sin(λ− β) ≈ sinλ− β cosλ,

cos(λ− β) ≈ cosλ+ β sinλ.

The magnetic field (2) is the field of rotating point-like magnetic dipole, while the electricfield (1) is a superposition of dipole and quadrupole fields. The quadrupole part is presentedby terms proportional to a2/ρ2 and decreases with distance as ρ−4. At great distances ρ� athis part vanishes and the electromagnetic field becomes that of rotating magnetic dipole.

The field near the surface of the magnetized body highly depends on the used model.The field (1) is calculated for a perfectly conducting sphere. The field of an inclined rotatorcalculated by authors cited above differs substantially from that given by Deutsch [7]. Butif we expand the field obtained for different models in powers of a, far from the surface ittakes the form of the field of a rotating point like dipole. For example, that is the case for

3

the fields which considered in [21], [20]. Hence, we consider the dipole field as a general caseat great distance. Then, the magnetic field is given by Eqs (2), and the electric field is

Er = 0,

Eθ = −µk3

ρ2C sinα, (3)

Eϕ =µk3

ρ2S sinα cos θ.

The fields (2) and (3) can be represented by 4-dimensional vector potential Aν . In thespherical coordinate system xν = (ct, r, θ, ϕ) it is

A0 = A1 = 0,

A2 = − µr3S sinα, (4)

A3 =µ

r3(cosα− C sinα cot θ).

In the next three sections we study dynamics and the potential energy for the chargedparticles in the field of rotating dipole given by Eqs (2) and (3), and in Section 6 we describethe potential energy near the sphere surface, according to Eqs (1) and (2).

3 Integral of motion for the particles in arbitrary ro-

tating electromagnetic field

Let us consider an arbitrary electromagnetic field rotating with angular velocity ω. Thefour-dimensional potential of such field in the inertial spherical coordinate system xν =(ct, r, θ, ϕ), ν = 0, 1, 2, 3 is defined as

Aν = Aν(r, θ, ϕ− ωt+ ρ).

In the corotating reference frame xν′

= (ct, r, θ, ψ), with ψ = ϕ − ωt, the field doesnot depend on time. Hence, the corresponding generalized momentum is conserved. TheLagrangian for a charged particle with mass m and charge e in rotating reference frame is

L =m

2uν

′uν′ +

e

cuν′A

ν′ , uν′= (ct, r, θ, ψ), (5)

where uν is the four-dimensional velocity, prime shows that the quantity relates to therotating reference frame, and the dot denotes derivative with respect to the proper time τ .As stated above, the time component p0′ of the generalized 4-momentum is an integral ofmotion:

p0′ =∂L

∂u0′= mu0′ +

e

cA0′ . (6)

This means that the energy of the particle in the corotating frame defined as E ′ = cp0′ isconserved.

4

Let us express the integral of motion P = p0′ in terms of quantities in the inertial referenceframe. The matrix of transformation from the inertial reference frame to the rotating onereads

J νµ′ =

∂xν

∂xµ′=

1 0 0 ω/c0 1 0 00 0 1 00 0 0 1

(7)

As a result of transformations Aµ′ = J νµ′ Aν we obtain

P = p0 +ω

cp3, (8)

where pν = muν + ecAν are the generalized momenta in the inertial frame. The quantity

p3 is the generalized angular momentum relative to the axis of rotation. If we multiply Eq.(8) by c, we can read it as: the sum of energy and angular momentum, multiplied by ω, isconserved.

Let us calculate the integral of motion for the field of precessing dipole. The generalizedmomenta calculated by use of Eqs (4) are:

p0 = mct, (9)

p3 = mϕ+eµ

cr3(cosα− C sinα cot θ). (10)

Using the metric tensor for the spherical coordinates in the inertial frame of reference

gµν = diag(1,−1,−r2,−r2 sin2 θ)

we find the integral of motion

P = m(ct− ω

cr2ϕ sin2 θ)− eµω

c2rsin θ(cosα sin θ − C sinα cos θ). (11)

Substituting ϕ = ψ + ωt we obtain the expression for P in the corotating reference system:

P = m[ct(1− ρ2 sin2 θ)− ωr2

cψ sin2 θ]− eµω

c2rsin θ{cosα sin θ − [cos(ρ+ ψ) + (12)

+ ρ sin(ρ+ ψ)] sinα cos θ}. (13)

If we consider r, θ, ψ as the particle coordinates, then the expression (12) is valid only insidethe light cylinder of radius c/ω, while Eq. (11) is correct throughout the entire space.

4 Potential energy

We study the particle dynamics with respect to the rotating reference frame with coordinatesct, r, θ, ψ and the metric defined by tensor

gµ′ν′ =

1− ρ2 sin2 θ 0 0 −rρ sin2 θ

0 −1 0 00 0 −r2 0

−rρ sin2 θ 0 0 −r2 sin2 θ

(14)

5

The total energy E ′ of a particle in curved space can be expressed as follows [22],

E ′ = cp0′ =mc2√g0′0′√

1− β2+ eA0′ , (15)

where β = v/c, and v is the particle velocity. As mc2√g0′0′ is the energy of the particle at

rest, we can define the kinetic energy as

T = mc2√g0′0′

(1√

1− β2− 1

)(16)

Then, the potential energy U can be introduced as U = cp0′ − T , which gives

U = mc2√g0′0′ + eA0′ (17)

The potential energy defined by Eq. (17) possesses a standard property: the space part ofits four dimensional gradient ∂νU is proportional to acceleration of the particle being at rest.In order to prove this we consider equation of motion

muν +e

cuσFσν −

m

2uσuρ∂νgσρ = 0, (18)

where Fσν = ∂σAν − ∂νAσ. Substituting the four-velocity uσ′

= (u0′, 0, 0, 0) for the particle

at rest we obtainmuν′ = u0

′(ec∂ν′A0′ +

m

2u0

′∂ν′g0′0′

),

It follows from uσ′uσ′ = c2 that u0

′= c/√g0′0′ . Hence,

muν′ =1

√g0′0′

∂ν′(eA0′ +mc2

√g0′0′

), (19)

which proves the statement.Let us find the potential energy of a particle in the rotating dipole field. Transformation

of the potential (4) into rotating reference frame by use of matrix (7) leads to

A0′ =µω

2rc[sinα sin 2θ(cos ξ + ρ sin ξ)− 2 cosα sin2 θ],

where ξ = ρ+ ψ. Substituting this into Eq. (17) and introducing a dimensionless potentialenergy V = U/mc2 we obtain

V =

√1− ρ2 sin2 θ +

N⊥2ρ

sin 2θ(cos ξ + ρ sin ξ)−N‖ρ

sin2 θ (20)

with

N⊥ = N sinα, N‖ = N cosα, N =eµω2

mc4. (21)

Notice, that all physical parameters are gathered in one dimensionless parameter N . Forexample, for electrons the value of N for pulsar in Crab Nebula is 5 · 1010, Jupiter – 0.03,Earth – 3 · 10−7 and for the magnetized sphere used in experiment by [23] – 3 · 10−16.

6

As shown in Paper I, in case of N � 1 the potential energy changes sufficiently in theregion ρ ∼ N1/3 � 1. Expanding (20) in power series in ρ we receive the effective potentialenergy studied in details in Paper I. It was shown there that the value of N plays the role ofa scale factor and in terms of reduced variable ρ = ρN−1/3 the shape of potential energy doesnot depend on N . In general case, the domain of potential energy is explicitly restricted bythe light cylinder ρ2 sin2 θ < 1, as indicated by Eq. (20).

5 Equipotential surfaces

In this section we present the profiles of potential energy defined by Eq. (20). Due to theargument ξ = ρ + ψ the equipotential surfaces take a form of surfaces twisted around theZ-axis. If we “twist back” the whole picture, introducing coordinate η = ψ + ρ− σ, with

sinσ =ρ√

1 + ρ2, cosσ =

1√1 + ρ2

. (22)

we find out that the potential energy becomes symmetric with respect to the plane η = 0, πwhich contains vectors µ and ω:

V =

√1− ρ2 sin2 θ +

N⊥2ρ

√1 + ρ2 sin 2θ cos η −

N‖ρ

sin2 θ. (23)

Besides, the function V is symmetric with respect to transformations η → η + π; θ → π− θ.At the plots below we show the sections of the equipotential energy surfaces by plane

η = const. The equipotential surfaces are marked by numbers equal to the energy levelV = const. It means that the particle having the total energy E ′ has zero velocity atequipotential surface V = E ′/mc2 and can move according to the equations of motion in thearea where the potential energy is less the its total energy. For example, a particle with totalenergy E ′ = 1.04mc2 being in the field depicted in Fig. 1 can move everywhere except theclosed region at the centre marked by number 1.04.

We study the structure of potential energy for α ≤ π/2, for positive and negative chargeof the particles. The structure for α > π/2 is the same but e should be replaced by −e andθ by π − θ. All profiles are plotted for the inclination angle α = 600, if other angle α is notspecified explicitly.

5.1 Equipotential surfaces for positively charged particles

We start with equipotential surfaces for small N . Fig. 1 and Fig. 2 show the profiles forN = 0.1. The sign of N is the sign of the particle charge according to definition of N (21).The equipotential surfaces for small N are almost the same as plotted in Paper I in 3-D form.For example, the equipotential surface for V = 0.74 is shown in Fig. 3. The constant C usedin Paper I and the energy level V of this paper are bound by relation V = 1 + N2/3C/2.The shape of equipotential surfaces at N � 1 does not depend on N . The value of N playsa role of scale factor in form V ∼ N2/3.

One can see in Figs. 1–3 that the energy levels form a potential valley in a shape of torusaround the centre of the field. There are two allowed regions for the particles of energy less

7

Figure 1: Equipotential profiles for N = 0.1, η = 0.

Figure 2: Equipotential profiles for N = 0.1, η = 900

8

than ≈ 0.74: one is the closed region inside the torus and another is the region outside thecylinder-like surface. At the critical energy level V ≈ 0.74 the inner and outer regions touchone another at two symmetric points which are the saddle points of the potential energy.This is demonstrated by Fig. 3. It was proved in Paper I that in approximation of small N

Figure 3: Equipotential surface for N = 0.1, V = 0.74

all the stationary points defined by equations ∂V/∂qi = 0 with qi = ρ, θ, ψ are the saddlepoints. There are also two saddle points in equatorial plane with coordinates defined byEq. (31). For the particles of energy greater than the critical energy, the inner and outerallowed regions are united by two symmetric conjugation tubes as one can see in Fig. 4.Such particles can escape from the torus-like trapping region to outer space.

Figure 4: Equipotential surface for N = 0.1, V = 0, 85

The potential profiles for intermediate values of N have similar structure. For example,the sections of equipotential surfaces for N = 1 are depicted in Figs 5–6. The saddle pointsfor this N lie at the energy level V ≈ −0.35. As N increases, the saddle points move tothe light cylinder along a lines given by Eq. (34). The line lying in the plane η = 0 is

9

Figure 5: Equipotential profiles for N = 1, η = 0

Figure 6: Equipotential profiles for N = 1, η = 900

Figure 7: Lines along which the saddle points moves as N varies. I – for negatively and II –for positively charged particles. η = 0, α = π/3.

10

shown in Fig. 7 as line II in the cylindrical coordinates R = ρ sin θ and Z = ρ cos θ. Thecorresponding lines in the plane η = π can be produced by substitution Z → −Z. It followsfrom Eqs (26–27) that the saddle points coordinates ρ sin θ → 1 as N →∞. I.e. the saddlepoints approach the light cylinder asymptotically when N grows. In other words, the criticalpotential surface, and hence, the trapping regions exist at any large N .

Figure 8: Equipotential profiles for N = 100, η = 0

Figure 9: Equipotential profiles for N = 100, η = 900

Sections of the potential surfaces for N = 100 are plotted in Fig 8 and Fig 9. The shapeof the profiles in case of large N does not depend on N . Indeed, if N � 1, we can neglect thefirst term in Eq. (23) and N becomes just a scale factor. The sole exception is the vicinityof the light cylinder, because the first derivatives ∂V/∂ρ and ∂V/∂θ, as one can see in Eqs(26) and (27), tend to infinity as ρ sin θ → 1.

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5.2 Equipotential surfaces for negatively charged particles

There is a significant difference between the structure of equipotential surfaces for positiveand negative charges, though they share a number of traits. As we change the sign of thecharge in expression for potential energy, the “potential hills” become “potential valley” andvice versa. The trapping regions in this case have a form of two symmetric dumb-bell shapedfigures. The sections of level surfaces for N = −1 are shown in Figs 10 and 11. For the

Figure 10: Equipotential profiles for N = −1, η = 0.

Figure 11: Equipotential profiles for N = −1, η = 900.

negatively charged particle there is still a critical energy level at which the inner trappingregion contacts the outer cylinder-like surface at two saddle points. The shape of the criticalsurface which is of level V ≈ 0.82 is depicted in Fig. 12.

As N varies, the saddle point in the plane η = 0 moves along the line I of Fig. 7 andapproaches the light cylinder of radius R = ρ sin θ = 1 as N →∞. The lines I and II in Fig.7 intersect at the coordinate origin at angle γ: cos γ = 1

3sinα.

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Figure 12: Equipotential surface for N = −0.1, V ≈ 0.82.

5.3 Orthogonal rotator

In this section we describe shortly the structure of potential energy for the inclination angleα = π/2. In this case the last term in Eq. (23) vanishes. Hence the equipotential surfacesfor positive and negative charges become symmetric because substitution e → −e is equalto substitution η → η + π.

The energy profiles for negative charge do not change sufficiently as α tends to π/2. Butthe profiles for positive charge change substantially as one can see in Figs 13 and 14.

The specific profiles of Fig. 14 arise as a consequence that the saddle points in equatorialplane, seen for example in Fig. 6, move to Z axis according to Eq. (31), as the angle αapproaches to π/2. And all the central pattern shrinks to the coordinate origin. The profilesfor negative charge are the same, but rotated around Z-axis through 1800. The trappingregions both for negative and positive charges form a dumb-bell shaped figures.

Figure 13: Equipotential profiles for N = 1, η = 0, α = π/2.

13

Figure 14: Equipotential profiles for N = 1, η = 900, α = π/2.

6 Potential energy near the surface of uniformly mag-

netized sphere

Up to now we have studied the particles dynamics on a large scale, up to the light cylin-der. The constructed plots are not valid near the surface of a star or a planet because thequadrupole electric field is neglected. As discussed in Sec. 2, the electric field not far fromthe surface essentially depends on the used model. In this section we investigate the poten-tial energy of charged particles in the field of perfectly conducting sphere found in [7]. Thesefields are described by Eqs (1) and (2). The respective 4-dimensional vector potential canbe written as follows

A0 = −ωr20µ

6cr3(3C sin 2θ sinα + cosα(3 cos 2θ + 1)) ,

A1 = 0,

A2 = − µr3S sinα, (24)

A3 =µ

r3 sin θ(cosα sin θ − C sinα cos θ),

Transforming this potential to rotating reference frame and substituting it into Eq. (17) weobtain the potential energy with regard to the quadrupole electric field

V =

√1− ρ2 sin2 θ +

[N⊥2ρ

sin 2θ(cos ξ + ρ sin ξ)−N‖ρ

sin2 θ

](1− a2

ρ2

)−

2N‖3ρ

a2

ρ2. (25)

It differs from the potential energy (20) by terms proportional to a2/ρ2.The magnitude of a for real celestial bodies is well below unity. For example, the values

of a for Earth, Jupiter and pulsar in Crab Nebula are 1.5 · 10−6, 4 · 10−5 and 7.6 · 10−3

respectively. We have plotted the equipotential surfaces for a = 0, 01. Built for the interval0 < ρ sin θ < 1, the graphs do not differ essentially from those presented in the previoussection, except for details at the coordinate origin. These details are shown in Figs 15 – 18.

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Figure 15: Equipotential profiles for N = 0.1, η = 0.

Figure 16: Equipotential profiles for N = 0.1, η = 900.

15

Figure 17: Equipotential profiles for N = −0.1, η = 0.

Figure 18: Equipotential profiles for N = −0.1, η = 900.

The distinctive property of potential energy in this case is that there is a minimum ofthe potential energy for negatively charged particles, and the trapping region is separatedfrom the surface of the sphere as shown in Fig. 17. The trapping region for the positivelycharged particles still forms of a torus encircling the star and siding the star surface alongthe magnetic equator (Figs 15 and 16). The shape of the profiles within the area ρ ∼ a doesnot change sufficiently as N varies. The reason is that the first term in Eq. (25) is closeto unity in case of small ρ, and it can be neglected as a constant in the potential energy.Hence, N becomes just a scale parameter.

7 Discussion

The potential energy of Eq. (23) was constructed under supposition that the magnetizedbody is in vacuum and there are not regions with non zero net charge in the surroundingplasma. But we see that the trapping regions for particles of different charge are located indifferent areas. Hence, this can cause sufficient charge separation in the magnetosphere. If

16

say, a pair of particles is born at some point as a result of pair production, one of the particlesfind itself in the potential well while the other one on the potential hill. As a consequencethe first one can be trapped in the region while the particle with opposite charge will leavethe region with acceleration. As the trapping regions accumulate sufficiently great charge,the potential profiles will be distorted. In this case we have to study the plasma dynamicsrather than motion of a single particle. A great variety of papers on this subject are citedin [8] and in recent review [24].

In the field with large N such as of neutron star, the relativistic charged particles undergointensive radiative friction force, leading to particle energy losses. Though, the particlemotion with respect to the radiation reaction is not a subject of this paper, the potentialenergy study is nevertheless, a powerful instrument of qualitative analysis of the particlebehaviour in this case. Because the structure of the potential energy is defined solely by thefield and does not depend on the particle motion. It is obvious that if the particle loses itsenergy due to radiation, it progressively passes to lower energy levels continuing its motionalong other magnetic field lines. Hence, the boundary restricting the allowed region changeswith time, shifting the particle downhill the potential profile. If the particle in this motionencounters the force-free surface defined by E ·H = 0, it can join the surface after someoscillations, and then drift along the surface as long as its energy is conserved [12, 25], butstill within the area bounded by appropriate equipotential surface. As may be seen fromthe figures above, the trapped positively charged particles eventually fall on the star surfaceas they lose their energy, while the others can move off to infinity. The particle orbitswith regard to radiation reaction have been numerically calculated in the field of orthogonalrotating dipole [26]. It was shown that there is a critical surface such that the trajectoriesstarting inside the surface end on polar regions, and the outside trajectories recede to infinity.

The fact that in case ω·µ > 0 the negatively charged particles concentrate in polar regionsof inclined rotator and the particles with positive charge in equatorial zone, coincides withconclusions made by other authors, who have used different models for the neutron starmagnetosphere [12,25,27].

Acknowledgement

This research has been supported by the grant for LRSS, project No 88.2014.2

A Stationary points of the relativistic potential energy

The power of potential formulation of the problem is the possibility to find the “potentialvalleys” where the charged particles can be trapped. And the slope of the “valley” showsthe force exerted on the particle. Having this in mind, we find the stationary points of thepotential energy, i.e. the points satisfying the set of equations:

∂V

∂qi= 0,

17

where qi = ρ, θ, ψ. This gives a system of three equations

ρ3 sin θ√1− ρ2 sin2 θ

+N⊥ cos θ√

1 + ρ2(cos η + ρ3 sin η)−N‖ sin θ = 0, (26)

ρ3 sin 2θ√1− ρ2 sin2 θ

− 2N⊥ cos 2θ√

1 + ρ2 cos η + 2N‖ sin 2θ = 0, (27)

sin 2θ sin η = 0. (28)

Equation (28) has two solutions:

i) θ =πn

2, n ∈ Z (29)

ii) η = 0, π. (30)

Solution i). The stationary points on the axis θ = 0, π can exist provided that ∂V/∂ρ = 0and ∂V/∂θ = 0 for any ψ, which is not the case as one can see in Eqs (26 – 27). As to the

equatorial plane θ =π

2, Eqs (26 – 27) have the next solutions:

ρ2 =N

2/3‖

21/3

3

√√√√1 +

√1 +

4N2‖

27+

3

√√√√1−

√1 +

4N2‖

27

; η = 0, π. (31)

Coordinate ρ in Eq. (31) increases monotone as N‖ increases, and asymptotically approaches

the value of unity as N‖ →∞. For small N‖ it takes the value ρ ≈ N1/3‖ .

Solution ii). Substituting η = 0, π into Eqs (26) and (27) we obtain two equations for ρand θ of the stationary points:

ρ3 sin θ√1− ρ2 sin2 θ

+εN⊥ cos θ√

1 + ρ2−N‖ sin θ = 0, (32)

ρ3 sin 2θ√1− ρ2 sin2 θ

− 2εN⊥ cos 2θ√

1 + ρ2 + 2N‖ sin 2θ = 0, (33)

where ε = 1 for η = 0 and ε = −1 for η = π. Solution of these equations gives the lines atwhich the stationary points are lying

tg θ = −ε3 cotα + q√

9 cot2 α + 8 + 4ρ2

2√

1 + ρ2, (34)

and equation for coordinate ρ at these lines:

ρ6Q2[4 + 4ρ2 +Q2]2 −N2 sin2 α[4 + 4ρ2 + (1− ρ2)Q2][2 +Q cotα]2 = 0, (35)

where Q = 3 cotα + q√

9 cot2 α + 8 + 4ρ2 and q = ±1 is the sign of the particle charge. IfN � 1 and ρ� 1 these equations transform to Eqs (36) and (37) of Paper I. The lines givenby Eq. (34) are plotted in Fig. 7 for α = π/3.

18

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